In all likelihood "modern" humans killed off the Neanderthals, keeping a few of the women for themselves, resulting in a slight infusion of Neandertal blood.
It is amazing how carefully scientists dance around this prospect, trying to pretend the correlation of time of exit of modern humans from Africa and extinction of the Neandertals is probably just a coincidence. The myth of the primeval Golden Age free from violence springs eternally new.
Since we know Neandertals were stronger than modern humans and this recent evidence implies they may also have been more intelligent, then the question arises as to how modern humans won the contest. Fighting a guy who is both smarter and stronger than you is seldom a good idea.
Humans out of Africa seemed to change spear-point styles over the ages as they tried new things - while Neanderthal humans seemed to be tied to the same styles.
They also may have been nice guys.
History has shown that we are not so nice, and not above pretending to be nice until the nice guys lower their guards.
Disease may have also played a role.
We may have also just out reproduced them. Humans have problems with reproduction due to the large skull, Neanderthals may have had even worse problems. It wouldn't take much of a difference to make a HUGE difference over time.
So what if, every time we come in conflict, they kill us two to one - if our population produces three healthy babies in the time it takes them to produce one.
I've been mugged by people smaller and likely less intelligent than me -- in groups of five or more.
If the Cro-Mags were, for example, faster or were able to throw further, then they could run up to the Neandertal camp, toss some rocks and spears at them, and run away from any response. Repeat until the Neandertals are all dead.